Managing Projects
Project Management Questions
What activities are required to complete
a project and in what sequence?
When should each activity be scheduled
to begin and end?
Which activities are critical to
completing the project on time?
What is the probability of meeting the
project completion due date?
How should resources be allocated to
activities?
Tennis Tournament Activities
ID
Activity Description
Network
Node
1 Negotiate for Location
A
2 Contact Seeded Players
B
3 Plan Promotion
C
4 Locate Officials
D
5 Send RSVP Invitations
E
6 Sign Player Contracts
F
7 Purchase Balls and Trophies
G
8 Negotiate Catering
H
9 Prepare Location
I
10 Tournament
J
Immediate
Predecessor
1
3
3
2,3
4
5,6
5,7
8,9
Duration
(days)
2
8
3
2
10
4
4
1
3
2
Notation for Critical Path Analysis
Item
Activity duration
Symbol
t
Definition
The expected duration of an activity
Early start
ES
The earliest time an activity can begin if all previous
activities are begun at their earliest times
Early finish
EF
The earliest time an activity can be completed if it
is started at its early start time
Late start
LS
The latest time an activity can begin without
delaying the completion of the project
Late finish
LF
The latest time an activity can be completed if it
is started at its latest start time
Total slack
TS
The amount of time an activity can be delayed
without delaying the completion of the project
Scheduling Formulas
ES = EFpredecessor (max)
(1)
EF = ES + t
(2)
LF = LSsuccessor
or
(min)
(3)
LS = LF - t
(4)
TS = LF - EF
(5)
TS = LS - ES
(6)
Tennis Tournament Activity on
Node Diagram
TS
A2
C3
START
B8
F4
D2
G4
E10
I3
H1
ES
EF
LS
LF
J2
Early Start Gantt Chart for Tennis
Tournament
ID
A
Activity
Days
2
C
Negotiate for
Location
Contact Seeded
Players
Plan Promotion
D
Locate Officials
2
E
I
Send RSVP
10
Invitations
Sign Player
4
Contracts
Purchase Balls
4
and Trophies
Negotiate
1
Catering
Prepare Location 3
J
Tournament
B
F
G
H
2
3
4
5 6
2
2
2
2
2 3
3
8
3
2
Personnel Required
Critical Path Activities
Activities with Slack
1
Day of Project Schedule
7 8 9 10 11 12 13 14 15 16 17 18 19 20
3
3
3
3
2
1
1
1
2
1
1
1 1
Resource Leveled Schedule for
Tennis Tournament
ID
A
Activity
Days
2
C
Negotiate for
Location
Contact Seeded
Players
Plan Promotion
D
Locate Officials
2
E
I
Send RSVP
10
Invitations
Sign Player
4
Contracts
Purchase Balls
4
and Trophies
Negotiate
1
Catering
Prepare Location 3
J
Tournament
B
F
G
H
2
3
4
5 6
Day of Project Schedule
7 8 9 10 11 12 13 14 15 16 17 18 19 20
2
2
2
2
2 2
2
8
3
2
Personnel Required
Critical Path Activities
Activities with Slack
1
2
2
2
2
2
2
3
2
2
2
2
1 1
Incorporating Uncertainty in
Activity times
F(D)
P(D<A) = .01
P(D>B) = .01
A
optimistic
M
most
likely
D
B
pessimistic
TIME
Formulas for Beta Distribution of
Activity Duration
Expected Duration
_
D
A4M  B
6
Variance
 B  A
V 

 6 
2
Note: (B - A )= Range or 6
Activity Means and Variances for
Tennis Tournament
Activity
A
B
C
D
E
F
G
H
I
J
A
1
5
2
1
6
2
1
1
2
2
M
2
8
3
2
9
4
3
1
2
2
B
3
11
4
3
18
6
11
1
8
2
D
V
Uncertainly Analysis
Assumptions
1. Use of Beta Distribution and Formulas For D and V
2. Activities Statistically Independent
3. Central Limit Theorem Applies ( Use “student t” if less than
30 activities on CP)
4. Use of Critical Path Activities Leading Into Event Node
Result
Project Completion Time Distribution is Normal With:
_
  D
For Critical Path Activities
 2  V
For Critical Path Activities
Completion Time Distribution for
Tennis Tournament
Critical Path
Activities
A
C
E
I
J
D
2
3
10
3
2
 = 20
V
4/36
4/36
144/36
36/36
0
2

188/36 = 5.2 =
Question
What is the probability of an overrun if a 24 day completion time
is promised?
Z
  52.
2
Z
X 

24  20
5.2
Z  175
.
24
P (Time > 24) = .5 - .4599 = .04 or 4%
Days
Costs for Hypothetical Project
Total Cost
Cost
Indirect Cost
Opportunity Cost
Direct Cost
(0,0)
Duration of Project
Schedule with Minimum Total Cost
Activity Cost-time Tradeoff
Cost
C*
Crash
Slope is cost to expedite per day
Normal
C
D*
D
Activity Duration (Days)
Cost-Time Estimates for Tennis
Tournament
Activity
A
B
C
D
E
F
G
H
I
J
Time Estimate
Normal Crash
2
1
8
6
3
2
2
1
10
6
4
3
4
3
1
1
3
2
2
1
Total
Direct Cost
Normal Crash
5
15
22
30
10
13
11
17
20
40
8
15
9
10
10
10
8
10
12
20
115
Expedite Cost
Slope
Progressive Crashing
Project
Duration
20
19
18
17
16
15
14
13
12
Activity
Crashed
Normal
Project Paths
A-C-D-G-I-J
A-C-E-I-J
A-C-E-H-J
A-C-F-H-J
B-F-H-J
Direct
Cost
115
Normal
Duration
16
20
18
12
15
Indirect
Cost
45
41
37
33
29
25
21
17
13
Opportunity
Cost
8
6
4
2
0
-2
-4
-6
-8
Duration After Crashing Activity
Total
Cost
168
Applying Theory of Constraints to
Project Management
 Why does activity safety time exist and is subsequently lost?
1. Dependencies between activities cause delays to
accumulate.
2. The “student syndrome” procrastination phenomena.
3. Multi-tasking muddles priorities.
 The “Critical Chain” is the longest sequence of dependent
activities and common resources.
 Replacing safety time with buffers
- Feeding buffer (FB) protects the critical chain from delays.
- Project buffer (PB) is a safety time added to the end of the
critical chain to protect the project completion date.
- Resource buffer (RB) ensures that resources (e.g. rental
equipment) are available to perform critical chain activities.
Accounting for Resource
Contention Using Feeding Buffer
NOTE: E and G cannot be performed simultaneously (same person)
A2
C3
START
D2
FB=7
G4
E10
I3
J2
FB=5
B8
F4
H1
Set feeding buffer (FB) to allow one day total slack
Project duration based on Critical Chain = 24 days
Incorporating Project Buffer
NOTE: Reduce by ½ all activity durations > 3 days to eliminate safety time
A2
C3
START
D2
E5
B4
F2
FB=2
G2
I3
H1
J2
PB=4
FB=3
Redefine Critical Chain = 17 days
Reset feeding buffer (FB) values
Project buffer (PB) = ½ (Original Critical Chain-Redefined Critical Chain)